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| Mirrors > Home > MPE Home > Th. List > 2moswapv | Structured version Visualization version GIF version | ||
| Description: A condition allowing to swap an existential quantifier and at at-most-one quantifier. Version of 2moswap 2644 with a disjoint variable condition, which does not require ax-13 2377. (Contributed by NM, 10-Apr-2004.) (Revised by GG, 22-Aug-2023.) Factor out common proof lines with moexexvw 2628. (Revised by Wolf Lammen, 2-Oct-2023.) |
| Ref | Expression |
|---|---|
| 2moswapv | ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nfe1 2150 | . . . 4 ⊢ Ⅎ𝑦∃𝑦𝜑 | |
| 2 | 1 | nfmov 2560 | . . . 4 ⊢ Ⅎ𝑦∃*𝑥∃𝑦𝜑 |
| 3 | nfe1 2150 | . . . . 5 ⊢ Ⅎ𝑥∃𝑥(∃𝑦𝜑 ∧ 𝜑) | |
| 4 | 3 | nfmov 2560 | . . . 4 ⊢ Ⅎ𝑥∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑) |
| 5 | 1, 2, 4 | moexexlem 2626 | . . 3 ⊢ ((∃*𝑥∃𝑦𝜑 ∧ ∀𝑥∃*𝑦𝜑) → ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑)) |
| 6 | 5 | expcom 413 | . 2 ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑))) |
| 7 | 19.8a 2181 | . . . . 5 ⊢ (𝜑 → ∃𝑦𝜑) | |
| 8 | 7 | pm4.71ri 560 | . . . 4 ⊢ (𝜑 ↔ (∃𝑦𝜑 ∧ 𝜑)) |
| 9 | 8 | exbii 1848 | . . 3 ⊢ (∃𝑥𝜑 ↔ ∃𝑥(∃𝑦𝜑 ∧ 𝜑)) |
| 10 | 9 | mobii 2548 | . 2 ⊢ (∃*𝑦∃𝑥𝜑 ↔ ∃*𝑦∃𝑥(∃𝑦𝜑 ∧ 𝜑)) |
| 11 | 6, 10 | imbitrrdi 252 | 1 ⊢ (∀𝑥∃*𝑦𝜑 → (∃*𝑥∃𝑦𝜑 → ∃*𝑦∃𝑥𝜑)) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∀wal 1538 ∃wex 1779 ∃*wmo 2538 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-10 2141 ax-11 2157 ax-12 2177 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-tru 1543 df-ex 1780 df-nf 1784 df-mo 2540 |
| This theorem is referenced by: 2euswapv 2630 2rmoswap 3767 |
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